Gain sure shouldn't be a problem in your system! I think the Wright phono pre had around 60dB of gain, which would get you 5V out of the phono pre with the Lyra and a 20:1 step-up. This is a ton of signal!
A 300B amp with 30dB of gain will make 8W at the speaker terminals with 0.25V coming into the amp itself. If use 10dB of feedback, then you need 0.8V at the input of the amp for 8W out. In either case, this is significant attenuation of the signal coming out of your phono preamp. Thankfully it has level controls!
Are you intending to replace the 300B amps and the Quickie with the Stereomour?
The Rel sub is actually an important piece of information to know about, as they generally have low-ish input impedances (10K or thereabouts) which become a factor if you're using a line level signal to drive them (as opposed to feeding them with speaker wire). It looks like Rel actually provides a cable with the Strata III for this purpose, and I'd certainly suggest using it!
There is also the question of how well the phono preamp can driver autoformers, which isn't super easy to answer without running some complicated tests.